Why this plan? The Beltrametti et al book is praised for stressing the geometric motivation of the algebraic machinery of AG. As someone who has studied some of the algebraic machinery without developing much intuition, this approach seems essential. Hartshorne is the canonical text, but Goertz and Wedhorn seems like a possible alternative. At this point, Eisenbud and Harris Geometry of Schemes would seem to be essential reading, although not as a primary text. Finally, my own research is in low-dimensional topology, hence my interest in complex surfaces, although it doesn't necessarily connect directly to my own research.

I should say something about the format of the reading group. Although I've studied some of this material already, I'm in no way qualified to teach it, and I'll be learning along with everyone else. The only way to learn this material is to do a shit ton of problems. Thus, we'll aim to do ALL the exercises. The main goal of the reading group is to help everyone stay motivated, set the pace, and provide a forum to discuss the material and problems in case people get stuck. My plan is to post notes I make when digesting the reading and solutions to problems. Everyone is encouraged to contribute similarly.

I'll aim to start May 21 and go on as long as we can/need. I'll have some time in the next two weeks to discuss the material if anyone wants to start before then. Also, it might be useful to review basic commutative algebra (rings, ideals, polynomial rings, etc.) before we get started, esp. if that material is new or rusty.

Finally, none of this is set in stone and if you have suggestions about the material or ideas about how to run the reading group, please post for everyone to discuss!

edit: I'm very excited by all the interest! I hope we can get some good momentum going. I created a subreddit for the reading group: r/AGreadinggroup.

Yes, several hours a day for months (or longer) is the plan. I'm a graduate student, this is what I do every day anyways.

Yes, I agree Vakil's notes seem like a good alternative to Hartshorne. I read Part I on categories and sheaves and some of the later stuff on derived categories. However, it is clear to me I personally have to start with a solid classical foundation.

I'm really interested in this, but I think my background is a bit sketchy. I have a lot of background in linear algebra and analysis (up through measure theory and functional analysis), but my knowledge of algebra is fairly patchy. I'm not terribly familiar with things like ideals and Noetherian rings.

I have a copy of Michael Artin's Algebra, and I've read up through Chapter 6. Any advice on what else I should read in this book to get up to speed for this reading group? It sounds like I have a few weeks to prepare.

Start with Chapter 10 (Rings), Chapter 13 (Fields) and then Chapter 12 (Modules). Next week I will probably have time to write up some background notes on ring and field theory for discussion. Also, if you have questions, please don't hesitate to post on the subreddit.

I'll definitely follow it but I may not be able to understand very much as I literally just finished an undergraduate course in abstract algebra and proof-based linear algebra (although I am trying to teach myself some things about modules and categories).

In fleshspace, as a grad course in a university in the middle east. I have only two students though. As for Shafarevich, I really think it has the best presentation of varieties from first principles, no scheme stuff in the first volume. I'm of the opinion that Hartshorne has the best exercises, but not the best presentation. Also, Joe Harris' book Algebraic Geometry has the best examples.

I'd avoid schemes, sheaves and cohomology in a first course in AG, and just try to develop intuition and an understanding of varieties. But if you do want to get into schemes and sheaves, one of my mentors really liked Mumford's Red Book.

You should be fine with second semester algebra. As long as you know the basic concepts from ring and field theory: rings, ideals, fields, field extensions, maximal and prime ideals, Noetherian rings, why R[x] is Noetherian, etc. Also, some basic point set topology would be good to know. We can review background as necessary, too.

Well, Im a physics/math double major, about to graduate, and I'm interested in doing physics theory (but I'm fine with learning math which is only tangentially related). So, I have done a lot on the physics side with manifolds (two general relativity classes) and am doing it a bit more rigorously in a math class now. Im also in a geometry class which is using Artin's Geometric Algebra, and Im really enjoying that.

I was thinking about teaching myself some functional analysis and homology, so I could study pde theory.

Sadly, I don't know enough physics to motivate AG from the physical perspective. Perhaps someone who knows more can weigh in. Here's a few other attempts at motivation:

Computational. We all know the Euclidean algorithm for computing gcd and Gaussian elimination for solving systems of linear equations. Turns out these have a rather elegant generalization called the theory of Groebner bases which give computational methods for solving systems of polynomial equations and other related problems.

Geometric. Varieties, the basic object of study in algebraic geometry, are sets of solutions to systems of polynomial equations. These are in a sense a generalization of manifolds. They can be smooth but also allow singular points. Many geometric concepts can be set up in this setting, and then generalized in surprising ways: for example, if you work over a finite field, you get arithmetic geometry. Your solution sets are finite points, but things like smoothness and tangent spaces still make sense!

Complex geometric. How compelling is complex geometry for you? If you work over the complex numbers, then complex AG and analytic (holomorphic) complex geometry are closely related. For example, every Riemann surface is a (smooth) complex algebraic curve.

Finally, I'll give one example where I know algebraic geometry shows up in mathematical physics. (This is something I would very much like to understand more, and part of my motivation for learning more AG.) Kontsevich's homological mirror symmetry conjecture states, loosely, that the Fukaya category of a symplectic manifold X is derived equivalent to the category of coherent sheaves on the mirror W, a complex algebraic variety.

Well, I cant say I really understood that last paragraph. However, your description of varieties interests me. I actually don't finish my quarter until June 9th, but I might try to catch up afterwards. We will see

I'd be very much up for this. Am a final year undergrad so have studied a couple of algebra and topology courses, I assume these will be enough as pre-requisites. I may be a bit quiet over the next couple of weeks whilst I take care of my exams, but this sounds great.

Do you plan on reviewing, at least briefly, the necessary concepts of algebra? I'm afraid my background in there is not enough, but I'd love to follow along. Nothing too fancy though, we could just recall the definitions. Any non-trivial theorems we need to assume?